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Optimal control and performance analysis of an MX/M/1 queue withbatches of negative customers

Published online by Cambridge University Press:  15 April 2004

Jesus R. Artalejo  and
Antonis Economou
Jesus R. Artalejo
Affiliation:
Department of Statistics and O.R., Faculty of Mathematics, Complutense University of Madrid, Madrid 28040, Spain; jesus_artalejo@mat.ucm.es.
Antonis Economou
Affiliation:
Department of Mathematics, University of Athens Panepistemiopolis, Athens 15784, Greece; aeconom@math.uoa.gr.
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Abstract

We consider a Markov decision process for an MX/M/1 queue that is controlled by batches of negative customers. More specifically, we derive conditions that imply threshold-type optimal policies, under either the total discounted cost criterion or the average cost criterion. The performance analysis of the model when it operates under a given threshold-type policy is also studied. We prove a stability condition and a complete stochastic comparison characterization for models operating under different thresholds. Exact and asymptotic results concerning the computation of the stationary distribution of the model are also derived.

Keywords

QueueingMarkov decision processesnegative customersstationary distributionstochastic comparison.
Type
Research Article
Information
RAIRO - Operations Research , Volume 38 , Issue 2: Advances in modelling of complex systems , April 2004 , pp. 121 - 151
Copyright
© EDP Sciences, 2004

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References

Artalejo, J.R., G-networks: A versatile approach for work removal in queueing networks. Eur. J. Oper. Res. 126 (2000) 233-249. CrossRef
D. Bertsekas, Dynamic Programming, Deterministic and Stochastic Models. Prentice-Hall, Englewood Cliffs, New Jersey (1987).
R.K. Deb, Optimal control of bulk queues with compound Poisson arrivals and batch service. Opsearch 21 (1984) 227-245.
Deb, R.K. and Serfozo, R.F., Optimal control of batch service queues. Adv. Appl. Prob. 5 (1973) 340-361. CrossRef
X. Chao, M. Miyazawa and M. Pinedo, Queueing Networks: Customers, Signals and Product Form Solutions. Wiley, Chichester (1999).
Economou, A., On the control of a compound immigration process through total catastrophes. Eur. J. Oper. Res. 147 (2003) 522-529. CrossRef
Federgruen, A. and Tijms, H.C., Computation of the stationary distribution of the queue size in an M/G/1 queueing system with variable service rate. J. Appl. Prob. 17 (1980) 515-522. CrossRef
Gelenbe, E., Random neural networks with negative and positive signals and product-form solutions. Neural Comput. 1 (1989) 502-510. CrossRef
Gelenbe, E., Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28 (1991) 656-663. CrossRef
Gelenbe, E., G-networks with signals and batch removal. Probab. Eng. Inf. Sci. 7 (1993) 335-342. CrossRef
Gelenbe, E., Glynn, P. and Sigman, K., Queues with negative arrivals. J. Appl. Prob. 28 (1991) 245-250. CrossRef
Gelenbe, E. and Schassberger, R., Stability of product form G-networks. Probab. Eng. Inf. Sci. 6 (1992) 271-276. CrossRef
E. Gelenbe and G. Pujolle, Introduction to Queueing Networks. Wiley, Chichester (1998).
Harrison, P.G. and Pitel, E., The M/G/1 queue with negative customers. Adv. Appl. Prob. 28 (1996) 540-566. CrossRef
O. Hernandez-Lerma and J. Lasserre, Discrete-time Markov Control Processes. Springer, New York (1996).
Kyriakidis, E.G., Optimal control of a truncated general immigration process through total catastrophes. J. Appl. Prob. 36 (1999) 461-472. CrossRef
Kyriakidis, E.G., Characterization of the optimal policy for the control of a simple immigration process through total catastrophes. Oper. Res. Letters 24 (1999) 245-248. CrossRef
T. Nishigaya, K. Mukumoto and A. Fukuda, An M/G/1 system with set-up time for server replacement. Transactions of the Institute of Electronics, Information and Communication Engineers J74-A-10 (1991) 1586-1593.
S. Nishimura and Y. Jiang, An M/G/1 vacation model with two service modes. Prob. Eng. Inform. Sci.  9 (1994) 355-374.
Nobel, R.D. and Tijms, H.C., Optimal control of an MX/G/1 queue with two service modes. Eur. J. Oper. Res. 113 (1999) 610-619. CrossRef
M. Puterman, Markov Decision Processes. Wiley, New York (1994).
S.M. Ross, Applied Probability Models with Optimization Applications. Holden-Day Inc., San Francisco (1970).
S.M. Ross, Introduction to Stochastic Dynamic Programming. Academic Press, New York (1983).
L.I. Sennott, Stochastic Dynamic Programming and the Control of Queueing Systems. Wiley, New York (1999).
Sennott, L.I., Humblet, P.A. and Tweedie, R.L., Mean drifts and the non-ergodicity of Markov chains. Oper. Res. 31 (1983) 783-789. CrossRef
Serfozo, R., An equivalence between continuous and discrete time Markov decision processes. Oper. Res. 27 (1979) 616-620. CrossRef
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983).
Teghem, J., Control of the service process in a queueing system. Eur. J. Oper. Res. 23 (1986) 141-158. CrossRef
H.C. Tijms, A First Course in Stochastic Models. Wiley, Chichester (2003).
Yang, W.S., Kim, J.D. and Chae, K.C., Analysis of M/G/1 stochastic clearing systems. Stochastic Anal. Appl. 20 (2002) 1083-1100. CrossRef